1. FIELD OF THE INVENTION
The present invention relates generally to optical transmission systems and more particularly to apparatus and methods for reducing the effect of nonlinear phase noise caused by the interaction of optical amplifier noise and Kerr effect.
2. Description of the Prior Art
For a system with many fiber spans having an optical amplifier in each span to compensate for fiber loss, a simplified mathematical description assumes that each optical amplifier launches the same average signal power and each fiber span has the same length. The launched signal, represented by baseband equivalent electrical field entering the first span is to E1=E0+n1, where E0 is the baseband representation of the transmitted electrical signal as a complex number and n1=x1+jy1 is the baseband representation of optical amplifier noise as zero-mean complex Gaussian noise with variance of E{|n1|2}=2σ2, where σ2 is the noise variance per span per dimension and E{.} denotes expectation. In the representation n1=x1+jy1, the quadrature components x1 and y1 are zero-mean real Gaussian noise with variances E{|x1|2}=E{|y1|2}=σ2. The simplest phase-modulated system uses binary phase shift keyed (BPSK) modulation where the binary “0” and “1” are modulated onto a carrier by a carrier phase shift of 0 or π, respectively. For a BPSK system, E0=+A or −A when “1” or “0” is transmitted, respectively, where A is a real number for the amplitude of the transmitted signal.
After the first in-line amplifier, the launched signal entering the second span is equal to E2=E0+n1+n2, where n2 is the amplifier noise from the optical amplifier of the second span. The statistical properties of n2 are the same as those of n1. At the end of amplifier chain after the Nth fiber span the signal becomes EN=E0+n1+ . . . +nN entering the Nth fiber span with noise from all N amplifiers, where nk, k=1 . . . N, is the noise from the kth fiber span, which has the same statistical properties as n1. For a simplified mathematical description, one may assume that the signal arriving at the receiver is equal to EN by ignoring the fiber loss of the last fiber span and the optical amplifier required to compensate for it.
FIG. 1 is an exemplary vector representation of a BPSK signal of “1” or “0” having constellation points with a phase of either 0 or π, respectively. Of course, any two phases that are separated by π may be used to represent “1” and “0”. The vector representation is shown on a complex plane having an x-axis 12, a y-axis 13, and an origin 14 at the intersection of the x and y axes 12 and 13. The signal “1” is transmitted as represented by a vector 15. A vector 16 represents amplifier noise of a complex number of n1+ . . . +nN in the absence of other effects. A vector 17 represents a received signal EN 15 with the effect of amplifier noise 16 without including an interaction of the amplifier noise and Kerr effect.
In this application the interaction of the amplifier noise and the Kerr effect is called Kerr effect phase noise. Gordon and Mollenauer in “Phase noise in photonic communications systems using linear amplifiers,” Optics Letters, vol. 15, no. 23, pp. 1351–1353, Dec. 1, 1990 describe the effects of the interaction of optical amplifier noise with Kerr effect in an optical fiber communication system. With the Kerr effect, the refractive index of an optical fiber increases linearly with the optical intensity in the fiber. In each span of optical fiber, the Kerr effect phase noise is equal to −γLeffP, where γ is the nonlinear coefficient of the optical fiber, Leff is the effective nonlinear length per span, and P is the optical intensity. The unit of electrical field is defined herein such that the optical intensity or power is equal to the absolute square of electric field. The nonlinear phase shift is usually represented by a negative phase shift in majority of the literature. The usage of positive phase shift does not change the physical meaning of the nonlinear phase shift. Herein, unless otherwise noted, the notation of negative phase shift is used.
A nonlinear phase noise is accumulated span after span due to the Kerr effect phase noise. The accumulated nonlinear phase noise is shown in an equation 1 below as nonlinear phase shift φNL, and denoted as an angle 18:φNL=−γLeff{E0+n1|2+|E0+n1+n2|2+ . . . +|E0n1+ . . . +nN|2}.  (1)The nonlinear phase shift φNL 18, which can also be represented by an electric field change vector 19, results in a received signal 20, which is the vector EN 17 rotated by the nonlinear phase shift φNL 18. Mathematically, the actual received electrical field 20 is ER=ENexp(jφNL). Because the actual received signal 20 is the rotated version of the signal plus noise vector 17, the intensity of the received signal 20 does not change due to nonlinear phase shift, i.e., PN=|EN|2=|ER|2.
In a long transmission system with many fiber spans, both the noise vector 16 and the nonlinear phase shift φNL 18 are accumulated span after span. The incremental nonlinear phase noise angle of the kth span is −γLeff|E0+n1+ . . . +nk|2 and is affected by the accumulated noise to that span of n1+ . . . +nk. Therefore, the nonlinear phase shift φNL 18 has a noisy component.
FIG. 2 shows a complex scattergram pattern, referred to with a reference 25, of a baseband representation of a received optical BPSK signal in the “1” state transmitted as E0=+A. The pattern 25 is plotted with 5000 noise simulation points. Each instance of the simulated complex field is graphed in the scattergram pattern 25 with real part as the x-axis 12 and imaginary part as the y-axis 13, and is represented as a single point according to the formula ENexp(jφNL−j<φNL>) which is equal to ERexp(−j<φNL>) with a mean nonlinear phase shift <φNL> of about 1 radian taken out. The points of the scattergram pattern 25 are similar to the end of the vector of 20 where the whole figure is rotated by minus the angle of the mean nonlinear phase shift <φNL>. The scattergram pattern 25 shows a random scattering of the simulation points due to contamination by the amplifier noise 16 and a shape having the appearance of a section of a helix due to the nonlinear phase shift of φNL−<φNL>. FIG. 2 indicates that the rotation of the nonlinear phase shift of φNL−<φNL> is correlated to the distance from the origin 14. In general, the farther the points are from the origin 14, the more the rotation about the origin 14.
Conventionally, as described widely in literature, the BPSK signal is detected by determining whether the signal is in the right or left of the y-axis 13, for “1” and “0”, respectively. A conventional detector viewed as a polar detector determines whether the absolute value of the received phase θ is less than π/2 for “1” or greater than π/2 for “0”. A conventional detector viewed as a rectangular detector determines whether one of the quadrature components, cos(θ), is positive or negative for “1” or “0”, respectively. In the conventional detectors, the intensity of the received signal need not be known in the detection process.
After taking out the mean <φNL> of the nonlinear phase shift φNL 18, Gordon and Mollenauer determine whether the signal is in the right or left of the y-axis 13 with the conventional phase detector. With the assumption of the conventional phase detector, Gordon and Mollenauer arrive at an estimation that the nonlinear phase shift mean <φNL> should be about less than 1 radian. This requirement of mean nonlinear phase shift <φNL> limits the transmission distance of an optical transmission system.
There is a need for a method and apparatus to overcome the interaction of optical amplifier noise and Kerr effect in order to extend transmission distance.